Geometry: An Interactive Journey to Mastery

The area of the inner square is

2 2 2 2

2

2

2

2

2

1 11

1

(^11)
(^1).
1
xab xx x
x
xx
x
x

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Lesson 29

1. %HFDXVHRSSRVLWHIDFHVRIDVWLFNRIEXWWHUDUHSDUDOOHODQ\FURVVVHFWLRQ
   that intersects a pair of opposite faces has a pair of parallel sides.
   Thus, all cross sections are parallelograms—or trapezoids if the slicing
   SODQHLQWHUFHSWVDVTXDUHHQGRIWKHEXWWHUDVVKRZQLQFigure S.29.1.
   ͼ௘$FWXDOO\FURVVVHFWLRQVFRXOGEHWULDQJXODUWRR'R\RXVHHKRZ"௘ͽ


2. ,WLV)ROORZSUHFLVHO\WKHVDPHDUJXPHQWGLVFXVVHGLQWKHOHVVRQJLYHQE\LQVHUWLQJWZRVSKHUHVLQWKH
   cylinder that each just touch the plane of the cross section.


3. 6XSSRVHWKDWDSRLQWP ͼ௘x, y௘ͽLVRQDQHOOLSVHZLWKIRFL
   F ͼ௘íc௘ͽDQGG ͼ௘c௘ͽVDWLVI\LQJFP + PG = k, for some
   QXPEHUk.1RWLFHE\WKHWULDQJXODULQHTXDOLW\WKDWFP + PG!
   FG. That is, k! 2 c6HHFigure S.29.2.
   The equation FP + PG = k reads
   xc y ^2222 xc y k.
   Rewrite this as
   xc y k  ^2222 xc y,
   DQGVTXDUHHDFKVLGHWRREWDLQ
   xc y k xc y  ^222222 2.k xc y 2

Figure S.29.1

P ͼx, yͽ

íC C

Figure S.29.2